Wikipedia
Izgoi is a term found in medieval Kievan Rus'. In primary documents, it is used to indicate orphans protected by the church. In historiographic writing on the period, it meant a prince in Kievan Rus' who was excluded from succession to the Kievan throne because his father had not held the throne before him.
In Kievan Rus', as well as Appanage and early Muscovite Russia, collateral succession, rather than linear succession was practiced, with the throne being passed from eldest brother to youngest brother and then to cousins down to the fourth in line of succession (not to be confused with "fourth cousins") in a generation before it was passed on to the eldest member of the senior line so long as his father had held the Kievan throne. The princes were placed in a hierarchy or "ladder" or "staircase" of principalities which Sergei Soloviev called the " Rota System" (rota being the Old Church Slavic term for a ladder or staircase), with Kiev as the pinnacle. When the grand prince of Kiev died, the next prince on the ladder moved up the ladder and the rest advanced a rung as well. Any prince whose father had not held the throne, say due to having predeceased the grandfather who was then grand prince, was excluded from succession and was known as izgoi.
The term is also found in the Expanded version of the Russkaya Pravda, in which case it meant an orphan or exile; thus an izgoi prince is in some sense seen as an "orphaned" or "exiled" prince in so far as he was left outside of the succession to the Kievan throne. But he was not, usually, landless, as is sometimes stated, as he still held the patrimonial land granted to him in the provinces.
A good example of an izgoi prince would be Vseslav of Polotsk, whose father, Briacheslav (d. 1044) and grandfather (Iziaslav d. 1020) both predeceased Vseslav's great grandfather, Vladimir the Great (d. 1015). Thus Vseslav was izgoi in so far as he could not legitimately claim the grand princely throne in Kiev since neither his father nor his grandfather had sat on the throne. He, however, remained prince of Polotsk, in Northeast Belarus. Furthermore, in spite of his excluded status, Vseslav did briefly seize the throne of Kiev in 1069, but held it only 6 months before being ousted. Another example (there are many others), would be Rostislav Vladimirovich, son of Vladimir Yaroslavich. Since Vladimir died in 1052, two years before his father, Yaroslav the Wise (d. 1054). Thus, since Vladimir had never held the Kievan throne, Rostislav was izgoi. His descendants, however, became princes of Galicia, in northwestern Ukraine. They were excluded from holding the grand princely throne in Kiev, but were not landless.